Spectral density of stochastic partial differential equations
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I am trying to understand some examples of computing the spectral density of SPDEs. The root of my confusion is that some terms appear to be missing from the calculation of the squared transform, and I can't work out why.
Example 1
First, here is an example I can follow. For the SPDE
$(kappa^2 + nabla cdot nabla)^{alpha/2} u(s) = mathcal{W}(s)$
we have
$mathbb{E}(|kappa^2 + nabla cdot nabla)^{alpha/2} hat{u}(s)|^2) = mathbb{E}(|hat{mathcal{W}}(s)|^2)$
and the spectral density is given by
$f(omega) = (2pi (kappa^2 + omega^2)^{alpha})^{-1}$.
Example 2
However, the next example I cannot understand. For the SPDE given by
$tau (varphi^2 - frac{partial^2}{partial t^2})^{1/2} u(s) = mathcal{W}(s)$
the spectral density is given as
$f(omega) = (tau 2pi (varphi^2 + omega^2))^{-1}$
but why is it not $tau^2$?
Example 3
Similarly, a later example considers the SPDE given by
$tau [rho frac{partial}{partial t} - Delta + kappa^2] u(s,t) = mathcal{W}(s,t)$
and the spectral density is given as
$f_{hd}(lambda, omega) = ((2pi)^{d+1} tau [rho^2omega^2 + (kappa^2 + lambda^2)^2])^{-1}$
whereas I would have expected to see more cross terms, e.g. $rho^2omega^2(kappa^2 + lambda^2)$.
functional-analysis differential-equations pde fourier-analysis stochastic-pde
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up vote
2
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I am trying to understand some examples of computing the spectral density of SPDEs. The root of my confusion is that some terms appear to be missing from the calculation of the squared transform, and I can't work out why.
Example 1
First, here is an example I can follow. For the SPDE
$(kappa^2 + nabla cdot nabla)^{alpha/2} u(s) = mathcal{W}(s)$
we have
$mathbb{E}(|kappa^2 + nabla cdot nabla)^{alpha/2} hat{u}(s)|^2) = mathbb{E}(|hat{mathcal{W}}(s)|^2)$
and the spectral density is given by
$f(omega) = (2pi (kappa^2 + omega^2)^{alpha})^{-1}$.
Example 2
However, the next example I cannot understand. For the SPDE given by
$tau (varphi^2 - frac{partial^2}{partial t^2})^{1/2} u(s) = mathcal{W}(s)$
the spectral density is given as
$f(omega) = (tau 2pi (varphi^2 + omega^2))^{-1}$
but why is it not $tau^2$?
Example 3
Similarly, a later example considers the SPDE given by
$tau [rho frac{partial}{partial t} - Delta + kappa^2] u(s,t) = mathcal{W}(s,t)$
and the spectral density is given as
$f_{hd}(lambda, omega) = ((2pi)^{d+1} tau [rho^2omega^2 + (kappa^2 + lambda^2)^2])^{-1}$
whereas I would have expected to see more cross terms, e.g. $rho^2omega^2(kappa^2 + lambda^2)$.
functional-analysis differential-equations pde fourier-analysis stochastic-pde
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I am trying to understand some examples of computing the spectral density of SPDEs. The root of my confusion is that some terms appear to be missing from the calculation of the squared transform, and I can't work out why.
Example 1
First, here is an example I can follow. For the SPDE
$(kappa^2 + nabla cdot nabla)^{alpha/2} u(s) = mathcal{W}(s)$
we have
$mathbb{E}(|kappa^2 + nabla cdot nabla)^{alpha/2} hat{u}(s)|^2) = mathbb{E}(|hat{mathcal{W}}(s)|^2)$
and the spectral density is given by
$f(omega) = (2pi (kappa^2 + omega^2)^{alpha})^{-1}$.
Example 2
However, the next example I cannot understand. For the SPDE given by
$tau (varphi^2 - frac{partial^2}{partial t^2})^{1/2} u(s) = mathcal{W}(s)$
the spectral density is given as
$f(omega) = (tau 2pi (varphi^2 + omega^2))^{-1}$
but why is it not $tau^2$?
Example 3
Similarly, a later example considers the SPDE given by
$tau [rho frac{partial}{partial t} - Delta + kappa^2] u(s,t) = mathcal{W}(s,t)$
and the spectral density is given as
$f_{hd}(lambda, omega) = ((2pi)^{d+1} tau [rho^2omega^2 + (kappa^2 + lambda^2)^2])^{-1}$
whereas I would have expected to see more cross terms, e.g. $rho^2omega^2(kappa^2 + lambda^2)$.
functional-analysis differential-equations pde fourier-analysis stochastic-pde
I am trying to understand some examples of computing the spectral density of SPDEs. The root of my confusion is that some terms appear to be missing from the calculation of the squared transform, and I can't work out why.
Example 1
First, here is an example I can follow. For the SPDE
$(kappa^2 + nabla cdot nabla)^{alpha/2} u(s) = mathcal{W}(s)$
we have
$mathbb{E}(|kappa^2 + nabla cdot nabla)^{alpha/2} hat{u}(s)|^2) = mathbb{E}(|hat{mathcal{W}}(s)|^2)$
and the spectral density is given by
$f(omega) = (2pi (kappa^2 + omega^2)^{alpha})^{-1}$.
Example 2
However, the next example I cannot understand. For the SPDE given by
$tau (varphi^2 - frac{partial^2}{partial t^2})^{1/2} u(s) = mathcal{W}(s)$
the spectral density is given as
$f(omega) = (tau 2pi (varphi^2 + omega^2))^{-1}$
but why is it not $tau^2$?
Example 3
Similarly, a later example considers the SPDE given by
$tau [rho frac{partial}{partial t} - Delta + kappa^2] u(s,t) = mathcal{W}(s,t)$
and the spectral density is given as
$f_{hd}(lambda, omega) = ((2pi)^{d+1} tau [rho^2omega^2 + (kappa^2 + lambda^2)^2])^{-1}$
whereas I would have expected to see more cross terms, e.g. $rho^2omega^2(kappa^2 + lambda^2)$.
functional-analysis differential-equations pde fourier-analysis stochastic-pde
functional-analysis differential-equations pde fourier-analysis stochastic-pde
edited Nov 14 at 11:00
asked Nov 13 at 12:42
prdnr
645
645
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1 Answer
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For example 3, I had missed that the modulus was being taken over complex numbers. The power spectral density is given by
$S(omega) = |hat{f}(i omega)^2| = hat{f}(i omega) hat{f}(-i omega)$
and in this particular case
$S(rho frac{partial}{partial t} - nabla + kappa^2) = (rho i omega + lambda^2 + kappa^2)(-rho i omega + lambda^2 + kappa^2) = rho^2 omega^2 + (lambda^2 + kappa^2)^2$
which gives the expected answer.
I'm still unsure about the $tau$ in example 2.
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
For example 3, I had missed that the modulus was being taken over complex numbers. The power spectral density is given by
$S(omega) = |hat{f}(i omega)^2| = hat{f}(i omega) hat{f}(-i omega)$
and in this particular case
$S(rho frac{partial}{partial t} - nabla + kappa^2) = (rho i omega + lambda^2 + kappa^2)(-rho i omega + lambda^2 + kappa^2) = rho^2 omega^2 + (lambda^2 + kappa^2)^2$
which gives the expected answer.
I'm still unsure about the $tau$ in example 2.
add a comment |
up vote
0
down vote
accepted
For example 3, I had missed that the modulus was being taken over complex numbers. The power spectral density is given by
$S(omega) = |hat{f}(i omega)^2| = hat{f}(i omega) hat{f}(-i omega)$
and in this particular case
$S(rho frac{partial}{partial t} - nabla + kappa^2) = (rho i omega + lambda^2 + kappa^2)(-rho i omega + lambda^2 + kappa^2) = rho^2 omega^2 + (lambda^2 + kappa^2)^2$
which gives the expected answer.
I'm still unsure about the $tau$ in example 2.
add a comment |
up vote
0
down vote
accepted
up vote
0
down vote
accepted
For example 3, I had missed that the modulus was being taken over complex numbers. The power spectral density is given by
$S(omega) = |hat{f}(i omega)^2| = hat{f}(i omega) hat{f}(-i omega)$
and in this particular case
$S(rho frac{partial}{partial t} - nabla + kappa^2) = (rho i omega + lambda^2 + kappa^2)(-rho i omega + lambda^2 + kappa^2) = rho^2 omega^2 + (lambda^2 + kappa^2)^2$
which gives the expected answer.
I'm still unsure about the $tau$ in example 2.
For example 3, I had missed that the modulus was being taken over complex numbers. The power spectral density is given by
$S(omega) = |hat{f}(i omega)^2| = hat{f}(i omega) hat{f}(-i omega)$
and in this particular case
$S(rho frac{partial}{partial t} - nabla + kappa^2) = (rho i omega + lambda^2 + kappa^2)(-rho i omega + lambda^2 + kappa^2) = rho^2 omega^2 + (lambda^2 + kappa^2)^2$
which gives the expected answer.
I'm still unsure about the $tau$ in example 2.
edited Nov 15 at 11:10
answered Nov 15 at 10:51
prdnr
645
645
add a comment |
add a comment |
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